The Sewol ferry sank on a routine trip south from the port of Incheon to the traditional holiday island of Jeju.
More than 300 people, most of them students and teachers on a field trip from the Danwon High School on the outskirts of Seoul, have died or are missing and presumed dead.
The children were told to stay put in their cabins, where they waited for further orders. The confirmed death toll on Sunday was 187.
The resignation has to be approved by President Park Geun-hye.
"Keeping my post too great a burden on the administration," a sombre Chung said in a brief announcement. "...On behalf of the government, I apologise for many problems from the prevention of the accident to the early handling of the disaster."
"There are too many irregularities and malpractices in parts of society that have been with us too long and I hope those are corrected so that accidents like this will not happen again."
Chung was booed and someone threw a water bottle at him when he visited grieving parents the day after the disaster.
Tempers have frayed over the slow pace of the recovery and frequent changes in information provided by the government.
The Gyeonggi Provincial Office of Education sent text messages to parents that "All Danwon High School students are rescued" in the hours after the disaster, media reported.
Park, who has the most power in government, was also booed by some of the relatives of the missing when she visited a gym where families of the missing were staying.